research article development of a low-level control system...

Research ArticleDevelopment of a Low-Level Control System for the ROV Visor3

Santiago Ruacutea and Rafael E Vaacutesquez

School of Engineering Universidad Pontificia Bolivariana Circular 1 No 70-01 050031 Medellın Colombia

Correspondence should be addressed to Rafael E Vasquez rafaelvasquezupbeduco

Received 23 February 2016 Revised 9 June 2016 Accepted 20 June 2016

Academic Editor Aleksandar Dogandzic

Copyright copy 2016 S Rua and R E Vasquez This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper addresses the development of the simulation of the low-level control system for the underwater remotely operatedvehicle Visor3 The 6-DOF mathematical model of Visor3 is presented using two coordinated systems Earth-fixed and body-fixedframes The navigation guidance and control (NGC) structure is divided into three layers the high level or the mission plannerthe mid-level or the path planner and the low level formed by the navigation and control systems The nonlinear model-basedobserver is developed using the extended Kalman filter (EKF) which uses the linearization of the model to estimate the currentstateThe behavior of the observer is verified through simulations using Simulink An experiment was conducted with a trajectorythat describes changes in the 119909 and 119910 and yaw components To accomplish this task two algorithms are compared a multiloopPID and PID with gravity compensation These controllers and the nonlinear observer are tested using the 6-DOF mathematicalmodel of Visor3 The control and navigation systems are a fundamental part of the low-level control system that will allow Visor3rsquosoperators to take advantage of more advanced vehiclersquos capabilities during inspection tasks of port facilities hydroelectric damsand oceanographic research

1 Introduction

Due to the growing interest around the world to performoffshore and underwater operations several researchers havefocused their interests on the construction of underwatervehicles that allow one to explore the ocean from a surfacestation Underwater vehicles are used to perform differ-ent tasks such as observation sampling and surveillanceRegardless of whether they are operated by cable (ROVs) orare autonomous (AUVs) it is necessary to develop controlstrategies to achieve the desired vehicle movements [1 2]

Several control schemes are based on the mathematicalmodel of the system Hence having accurate models for pre-diction and control is desirable however this is not a simpletask due to the highly nonlinear behavior that appears withthe fluid-vehicle interaction [3] The navigation guidanceand control (NGC) system for an underwater vehicle can havedifferent degrees of sophistication depending on the type ofoperation that is to be performed and the autonomy levelsthat need to be achieved [1 2 4 5]

The desired level of autonomy will determine what kindsof algorithms are necessary to control the variables of interest

which are normally given by the position attitude (orienta-tion) and velocity of the vehicle with respect to an inertialreference system located at the surface [6] Figure 1 showsa three-level hierarchical NGC structure for an underwatervehicle this kind of structure is useful to control and stabilizethe vehicle [7]

One of the main components in the control structurersquoslower level is the navigation system This system provides anestimate of the position velocity and attitude of the vehiclewith respect to an inertial system located at the surfacecontrol station at the harbor from measurements made withdifferent sensors (IMU magnetometer depth DVL andSSBL among others) Given the characteristics of water thedevelopment of underwater localization systems is not trivialand presents a number of challenges [8 9]

The most common algorithm to achieve this task is theKalman filter (KF) It is an estimator statistically optimal withrespect to a quadratic error function which allows one toestimate the state of the vehicle [10] Several Kalman-filter-based navigation systems have been developed for yearsand all of them depend on the instrumentation used in

Hindawi Publishing CorporationInternational Journal of Navigation and ObservationVolume 2016 Article ID 8029124 12 pageshttpdxdoiorg10115520168029124

2 International Journal of Navigation and Observation

High level

Mid-level

Low level

Mission

Mission planning

Waypoints

Path planning

Setpoints

Control Vehicle

Position

Sensors

Navigationsystem

velocityattitude

Figure 1 Control structure for an underwater vehicle

the vehicle Caccia et al [11] developed a Kalman-filter-based acoustic navigationmodule to control theUUVRoby2Caccia and Veruggio [12] used Kalman-filter techniques toestimate the state using different sampling rates of a depth-meter and altimeter Drolet et al [13] presented an integratedsensor fusion strategy using multiple Kalman filters allowingdifferent combination of sensors Blain et al [14] developedand tested a Kalman filter to merge data from an acousticpositioning system a bathymeter and a DVL Loebis et al[15] implemented an intelligent navigation system based onthe integrated use of the global positioning system (GPS)and several inertial navigation system (INS) sensors forAUV Kinsey et al [16] presented a survey with advances inunderwater vehicle navigation and identified future researchchallenges Lee and Jun [17] presented a pseudo long baseline(LBL) navigation algorithm using the EKF Watanabe et al[18] proposed an accurate tracking method to estimate AUVposition by using a super-short baseline (SSBL) from theship Geng and Sousa [19] presented a hybrid derivative-free extended Kalman filter taking advantage of both thelinear time propagation of the Kalman filter and nonlinearmeasurement propagation of the derivative-free extendedKalman filter

The control system is another component of the lowerlevel of the control structure It contains a set of algorithmsthat stabilize the state of the vehicle so it can follow thecommands generated in the path planning system Thecontrol of an underwater vehicle is complex because thereare highly nonlinear hydrodynamic effects resulting from theinteraction with the environment that cannot be quantified[20] Cohan [21] states that the development of controlsystems for ROVs is a current and promising topic for futuredevelopments this can be verified with the number of papersthat can be found in literature

Caccia and Veruggio [12] implemented and tested aguidance and control system for underwater vehicles usingprogrammed controllers to regulate speed at the low level ina hierarchical three-level structure Do et al [22] developeda robust adaptive control strategy to ensure that a six-degree-of-freedom vehicle follows a prescribed path usingfour actuators Van de Ven et al [23] presented a qualitativeassessment of the performance of control strategies usingneural networks indicating the advantages disadvantagesand application recommendations Hoang and Kreuzer [24]

designed an adaptive PD controller for dynamic positioningof ROVs when the mission is executed in places nearsubmerged structures and requires great execution precisionBessa et al [25] used slidingmode controllers combinedwithfuzzy adaptive algorithms for controlling depth in ROVsAlvarez et al [26] developed a robust PID controller forcontrolling AUV used in oceanographic sampling workSubudhi et al [27] presented the design of a feedbackcontroller for tracking paths in vertical planes Ishaque etal [28] presented a simplification of the conventional fuzzycontroller for an underwater vehicle Herman [29] presenteda decoupled PD setpoint controller which is expressed interms of quasi-velocities for underwater vehicles Petrich andStilwell [30] presented a robust control for an autonomousunderwater vehicle that suppresses pitch and yaw coupling

Gutierrez et al [31] developed the underwater remotelyoperated vehicle Visor3 for surveillance and maintenanceof ship hulls and underwater structures of Colombian portfacilities and oceanographic research Visor3 has a three-layerhardware architecture instrumentation communicationsand control [32] Although much work has been done inmechanics and electronics a closed-loop control system wasnot developed for the ROV Visor3 so the capabilities of thevehicle are still completely dependent on the pilot skills

This work addresses the first approach to develop thelow-level control system for the ROV Visor3 that will helpoperators to maneuver and determine the position andattitude of the vehicle under certain scenarios such as in portsor dams inspection tasks The second section presents themathematical model of the vehicle the third section showsthe development of the nonlinear model-based observerand the fourth section presents the implemented controlalgorithms then the simulation results are shown and someconclusions are presented

2 Mathematical Model

To analyze themotion ofVisor3 in a three-dimensional spacetwo coordinate frames are defined an inertial Earth-fixedframe (this reference frame is considered as the North-East-Down (NED) frame attached to a port facility) where themotion of the vehicle is described and a body-fixed framewhich is conveniently fixed to the vehicle and moves withit Figure 2 The acceleration of the Earth due to rotation isneglected for this work The position and orientation of thevehicle are described relative to the Earth-fixed frame as

120578 = [119909 119910 119911 120601 120579 120595]T (1)

where 1205781= [119909 119910 119911]

T is the position and 1205782= [120601 120579 120595]

T

is the orientation The linear and angular velocities of thevehicle relative to the body-fixed frame are given by

^ = [119906 V 119908 119901 119902 119903]T (2)

where ^1= [119906 V 119908]T and ^

2= [119901 119902 119903]

T are the linear andangular velocities respectively

International Journal of Navigation and Observation 3

Y0 Y

X0

Incoming-flow Z0

Body-fixed

r0

X

Z

Earth-fixed

Xf

minus120573

120572

(sway)q (pitch)

p (roll)

u (surge)

r (yaw)

w (heave)

Figure 2 Coordinate frames and velocities (linear and angularcomponents)

The mathematical model that describes the 6-DOF dif-ferential nonlinear equation of motion for an underwatervehicle stated in [6] is given by

M ^ + C (^) ^ +D (^) ^ + g (120578) = 120591

= J (120578) ^(3)

where M isin R6times6 is the inertia matrix which comprises themass of the rigid body and the added mass M = MRB +

M119860 C isin R6times6 is the Coriolis and centripetal matrix which

includes a term due to rigid body and a term due to the addedmass C = CRB + C

119860 D isin R6times6 is the damping matrix

g isin R6times1 is the gravitational andmoments vector 120591 isin R6times1 isthe force vector and J isin R6times6 is the rotation matrix from thebody-fixed frame to the Earth-fixed frame For this work it isassumed that the origin of the body-fixed frame is located atthe same point of the center of gravity of the vehicle

Applying Newtonian laws of motion the rigid bodyequation of motion for the vehicle is given by

MRBk + CRB (k) k = 120591RB (4)

In (4) the rigid body inertia matrixMRB can be expressed as

MRB = [119898I3times3

minus119898S (r119866)

119898S (r119866) I

0

] (5)

where119898 is the mass of the vehicle I3times3

is the identity matrixI0is the inertia tensor with respect to the center of gravity

r119866is the gravity vector in the body-fixed frame and S(sdot) is a

skew symmetric matrix The centripetal and Coriolis matrixcan be parametrized in the form of a skew symmetric matrixas follows

CRB = [CRB11

CRB12

CRB21

CRB22

] (6)

whereCRB11

= 03times3

CRB12

= minus119898S (^1) minus 119898S (^

2) S (r119866)

CRB21

= minus119898S (^1) + 119898S (^

2) S (r119866)

CRB22

= minusS (I0) ^2

(7)

The added mass due to the inertia of the surrounding fluid isgiven by

M119860= [

A11

3times3 A12

3times3

A21

3times3 A22

3times3]

= minus[

120597120591

120597

120597120591

120597V120597120591

120597

120597120591

120597

120597120591

120597

120597120591

120597 119903

]

(8)

where 120591 = [119883 119884 119885 119870 119872 119873] are the hydrodynamicsadded-mass forces and moments in each direction Findingthe 36 entries of M

119860is a difficult task but this can be

simplified by using symmetry properties of the vehicle Forthis work Visor3 is assumed to be symmetric with respect to119909119910 119909119911 and 119910119911 planes Therefore the added-mass matrix canbe computed as

M119860= minus diag 119883

119884V 119885 119870119872 119873 119903 (9)

The hydrodynamic centripetal and Coriolis matrix can alsobe parametrized as

C119860(^) = [

03times3

minusS (A11^1+ A12^2)

minusS (A11^1+ A12^2) minusS (A

21^1+ A22^2)

] (10)

For underwater vehicles damping is mainly caused by skinfriction and vortex shedding One approximation commonlyused [6 33 34] is a term with linear and quadratic compo-nents given by

D (^) = minus diag 119883119906 119884V 119885119908 119870119901119872119902 119873119903

minus diag 119883119906|119906| |119906| 119884V|V| |V| 119885119908|119908| |119908| 119870119901|119901|

10038161003816100381610038161199011003816100381610038161003816

119872119902|119902|

10038161003816100381610038161199021003816100381610038161003816 119873119903|119903| |119903|

(11)

Restoring forces and moments calculated from center ofgravity are given by

g (120578) =

[

[

[

[

[

[

[

[

[

[

[

[

(119882 minus 119861) sin 120579minus (119882 minus 119861) cos 120579 sin120601minus (119882 minus 119861) cos 120579 cos120601

119910119887119861 cos 120579 cos120601 minus 119911

119887119861 cos 120579 sin120601

minus119911119887119861 sin 120579 minus 119909

119887119861 cos 120579 cos120601

119909119887119861 cos 120579 sin120601 + 119910

119887119861 sin 120579

]

]

]

]

]

]

]

]

]

]

]

]

(12)

where119882 = 119898119892 is the weight of the vehicle and 119861 = 120588119892nabla isthe buoyancy 120588 is the density of the fluid 119892 is the accelerationof the gravity and nabla is the volume of fluid displaced by thevehicle


21 Transformations To obtain the linear velocity in theEarth-fixed frame from the linear velocity in the body-fixed frame a linear transformation must be applied Thetransformation matrix is given by

J1(1205782)

=[

[

[

c120595c120579 minuss120595c120601 + c120595s120579s120601 s120595s120601 + c120595s120579c120601s120595c120579 c120595c120601 + s120595s120579s120601 minusc120595s120601 + s120595s120579c120601minuss120579 c120579s120601 c120579c120601

]

]

]

(13)

where ssdot = sin(sdot) and csdot = cos(sdot) The angular velocitytransformation matrix is given by

J2(1205782) =

[

[

[

[

[

1 s120601t120579 c120601t1205790 c120601 minuss120601

0

s120601c120579

c120601c120579

]

]

]

]

]

(14)

where tsdot = tan(sdot)Finally the kinematic equations of the vehicle can be

expressed as

[

1

2

] = [

J1(1205782) 03times3

03times3

J2(1205782)

] [

^1

^2

] (15)

3 Nonlinear Model-Based Observer

The development of the observer needs to account for theROV high nonlinearities and coupled dynamics Thereforean extended 6-DOF Kalman filter (EKF) that considersCoriolis damping and restoring forces has been developedEquations (3) can be written as a state-space realization asfollows

x = 119891 (x 119905) + Bk (119905) +m (119905) (16)

z = ℎ (x 119905) + n (119905) (17)

where 119891(x 119905) is a function of the state vector x = [^ 120578]T andis given by

119891 (x 119905) = [Mminus1 (minusC (^) ^ minusD (^) ^ minus g (120578))

J (120578) ^] (18)

and B is the input coupling matrix given by

B = [Mminus1

06times6

] (19)

In (16) k(119905) is the input vector given by thrustersrsquo forcesℎ(x 119905) is the measurement sensitivity matrix which dependson the ROV sensors and m(119905) and n(119905) are plant andmeasurement noises respectively they are assumed to be

zero-mean Gaussian white noise processes with covariancematrixQ(119905) and R(119905) given by

119864 ⟨m (119905)⟩ = 0

119864 ⟨m (119905)mT(119904)⟩ = 120575 (119905 minus 119904)Q (119905)

119864 ⟨n (119905)⟩ = 0

119864 ⟨n (119905)nT(119904)⟩ = 120575 (119905 minus 119904)R (119905)

(20)

The continuous-time model in (16) and (17) is discretizedusing a 1st-order approximation Euler method as follows

x119896= 119892119896minus1(x119896minus1) + Δk

119896minus1+m119896minus1

z119896= ℎ119896(x119896) + n119896

(21)

where

119892119896minus1(x119896minus1) = x119896minus1+ ℎ119891 (x

119896minus1 119905119896minus1)

Δ = ℎB(22)

and ℎ is the step time The discretized system is given by

^119896= ^119896minus1+ ℎMminus1 [120591

119896minus1minus C (^

119896minus1) ^119896minus1

minusD (^119896minus1) ^119896minus1minus g (120578

119896minus1)]

120578119896= 120578119896minus1+ ℎ [J (120578

119896minus1) ^119896]

(23)

The objective of the EKF is to estimate the current stateby using a linearized version of the systemrsquos model TheEKF is executed in two steps the predictor which calculatesan approximation of the state and covariance matrix andthe corrector which improves the initial approximation Thepredictor equations for the EKF are

x119896(minus) = 119892

119896minus1(x119896minus1(+)) + Δk

119896minus1

z119896= ℎ119896(x119896(minus))

P119896(minus) = Φ

119896minus1P119896minus1(+)Φ

T119896minus1+Q119896minus1

(24)

x119896(minus) is the a priori estimate of the state x

119896(+) is the a

posteriori estimate of the state z119896is the predicted measure-

ment P119896(minus) is the a priori covariance matrix P

119896(+) is the a

posteriori covariance matrix and Φ119896minus1

is the state transitionmatrix defined by the following Jacobian matrix

Φ119896minus1asymp

120597119892119896

120597x

10038161003816100381610038161003816100381610038161003816x=x119896minus1(minus)

(25)

The corrector equations for the EKF algorithm are

x119896(+) = x

119896(minus) + K

119896(z119896minus z119896)

P119896(+) = [I minus K

119896H119896]P119896(minus)

(26)

where K119896is the Kalman-filter gain calculated as

K119896= P119896(minus)HT

119896[H119896P119896(minus)HT

119896+ R119896]

minus1

(27)


Open-loop control system ROV dynamics ROV kinematics

Environmentalperturbations

Pilot Joystick B+

minus minus

usum

g

C + D

Jint intMminus1Thrusterallocation

120578120591 ^

Figure 3 Open-loop control system implemented in Visor3

The observation matrix is defined by the following Jacobianmatrix

H119896asymp

120597ℎ119896

120597x

10038161003816100381610038161003816100381610038161003816x=x119896(minus)

(28)

It is important to state that the EKF is not an optimal filterdue to the linearization process of the system Furthermorethe matrices Φ

119896minus1and H

119896depend on the previous state

estimation and the measurement noise Therefore the EKFmay diverge if consecutive linearizations are not a goodapproximation of the linear model in the whole domainHowever with good knowledge of the systemrsquos dynamics biasand noise compensation and an appropriate configurationsampling time this algorithm is good enough for applicationin control systems in marine vehicles

4 Control Algorithms

Thecontrol of an underwater vehicle is complex because thereare highly nonlinear hydrodynamic effects resulting from theinteraction with the environment that cannot be quantified[20] Additionally disturbances from the environment mayappear Problems such as obtaining all the state variables forthe vehicle can limit the design of such control algorithms

Currently the ROV Visor3 is driven through a joystickthat sends power commands to a main-board installed in thevehicle this board translates commands into input signals foreach motor Figure 3 shows the current open-loop controlsystem implemented in Visor3

41 Thruster Allocation The task which consists in generat-ing a particular command to be sent to each individual actu-ator according to the control law and thrusters configurationis called control allocation [33]

The thrust vector that describes the force 119891119894generated by

the thruster 119894 is given by

f = [1198911 1198912 sdot sdot sdot 119891119903]T (29)

where 119903 is the total number of thrusters The total force andmoment generated by the thrusters will be

120591 = Tf (30)

where T is the thruster configuration matrix which is afunction of the thrusters position r119887

119905119894119887 as well as the azimuth

and elevation angles 120572 and 120573 respectively This vector is ref-erenced to the body-fixed frame The thruster configurationmatrix describes how the thrust of each motor contributes tothe force or moment in each direction The matrix T is givenby

T = [t1 t2sdot sdot sdot t119903] (31)

where t119894is the column vector of the 119894th thruster and is

computed as

t119894= [

I3times3

minusST (r119887119905119894119887)

]R (120572 120573)[[[

1

0

0

]

]

]

119891119894 (32)

The rotation matrix R(120572 120573) is defined by the product of tworotation matrices

R (120572 120573) = R119911120572R119910120573 (33)

where R119911120572

and R119910120573

are described by

R119911120572=[

[

[

c120572 minuss120572 0s120572 c120572 00 0 1

]

]

]

(34)

R119910120573=[

[

[

c120573 0 s1205730 1 0

minuss120573 0 c120573

]

]

]

(35)

respectively Visor3 has fixed heading and pitch angles foreach thruster


Now that the thrust provided by each thruster is knownusing (30) the input that must be sent to each motor has tobe computed The input could be the revolution speed of themotor or a voltage signal for the driver Equation (30) is thenrewritten as

120591 = TKu (36)

where K = diag 1198701 1198702 sdot sdot sdot 119870119903 is a diagonal matrix withthe thrust coefficients described by the equation

119870119894= 119870119879(119869) 120588119863

4 (37)

u is a column vector with elements 119906119894= |119899|119899 where 119899 is the

propeller velocity rate 120588 is the water density and 119863 is thepropeller diameter The thruster allocation problem is solvedfinding u as

u = Kminus1Tminus1120591 (38)

Although Tmay not be square or invertible it can be solvedusing the Moore-Penrose pseudo inverse given by

Tdagger = TT(TTT

)

minus1

(39)

Substituting (38) into (39) yields

u = Kminus1Tdagger120591 (40)

The voltage for each driver calculated from u is given by

V119894= (

1

119866119898119894

)(

1

119866119889119894

) sgn (119906119894)radic1003816100381610038161003816119906119894

1003816100381610038161003816 (41)

where 119866119898119894

and 119866119889119894

are the gain of the 119894th motor anddriver respectively This last equation fails when saturationoccurs in the actuators Figure 4 shows how to change V

119894

according to 119906119894 with different values from multiplication

119872119892= (1119866

119898119894

)(1119866119889119894

)

42 Multivariable PID Control The parallel noninteractingstructure of the PID is given by

120591PID = K119901e (119905) + K119889e (119905) + K119894 int119905

0

e (120591) 119889120591 (42)

where K119901is the proportional gain K

119889is the derivative gain

K119894is the integral gain and e(119905) is the error defined by

e = 120578119889minus 120578 (43)

For Visor3 each controller is designed for the control ofone DOF this implies that K

119901 K119894 and K

119889are diagonal and

positive matrices In this work heuristic methods were usedto tune the gains of the PID controller A high proportionalgain acts rapidly to correct changes in the references Dueto the vehiclersquos dynamics and the interaction with the fluida high derivative action is used to provide damping to themotion of the vehicle when it is reaching the setpoint Finallyit is decided that a small integral action can correct the steady-state error This PID control can be improved according

i(V

)

5 10 15 200ui (rps)

Mg = 01

Mg = 03

Mg = 05

Mg = 07

Mg = 09

Mg = 11

0

1

2

3

4

5

Figure 4 Change of voltage according to 119906119894in the thruster

allocation problem

to Fossen [6] by using gravity compensation and vehiclersquoskinematics The PID control signal is transformed by

120591 = JT (120578) 120591PID + g (120578) (44)

Figure 5 shows the implementation of (44) The controllerneeds the error and the position estimation to calculate theoutput

5 Simulation and Results

The simulation of the ROV dynamics the observer algo-rithm and the controller were implemented in SimulinkAdditionally several functions can be constructed usingconventional MATLAB code providing great flexibility forhigh-level programming

Visor3 parameters were obtained using CAD models(Solid-Edge software) and CFD simulation (ANSYS soft-ware) Table 1 contains all the model parameters used in thesimulation to represent the dynamics of the vehicle

The thruster simulation is divided into three steps thedriver the motor dynamics and the propeller dynamics Thelow-level control of the system is managed by the driver Thedriver regulates the torque by increasing or decreasing thecurrent in order to maintain the velocity of the motor shaftFor this work it is assumed that load changes generated forthe propellers are regulated by the driver In that order onlythrust is considered

For this work motorsrsquo dynamics are not considered sincethey present a fast time response in comparison with thedynamics of the vehicle Figure 6 shows the dynamic responseof the MAXON DC motor The steady-state value is reachedin less than 40ms The simulation model is represented only


Closed-loop control system ROV dynamics ROV kinematics

PilotJoystick Filter PID

Sensors


B+

minus

+

minus minussumsumsum Mminus1

uJint int

C + DJT(middot)

g(middot)

g(middot)

Tdagger120578p 120578d 120578

times

^

Figure 5 Closed-loop control system implemented in Visor3

Table 1 ROV Visor3 parameters for simulation

Parameter Value119898 645 kg119868119909119909

29 kgm2

119868119910119910

25 kgm2

119868119911119911

30 kgm2

119868119909119910

minus70 times 10minus3 kgm2

119868119909119911


119868119910119911


nabla 18 times 10minus2m3

[119909119892 119910119892 119911119892] [0 0 0]m

[119909119887 119910119887 119911119887] [17 18 68] times 10

minus3m119883

65 kg119884V 598 kg119885

598 kg119870

0 kgm2

119872

22 kgm2

119873119903

22 kgm2

119883119906|119906|

minus103 kgm119884V|V| minus1008 kgm119885119908|119908|

minus1008 kgm119870119901|119901|

minus4003 kgm2

119872119902|119902|

minus1008 kgm2

119873119903|119903|

minus1008 kgm2

by a constant gain 119866119898 which is the relationship between the

maximum velocity Velmax and the nominal voltage 119881nom andis described by

119866119898=

Velmax119881nom

(45)

The motor used is a MAXON EC motor 136201 with aplanetary gearhead GP 42C-203113 The nominal speedreduction and the nominal voltage are taken from [35 36]and used in (45) to obtain 119866

119898

0

200

400

600

800

1000

1200

120596(t)

(rpm

)

005 01 015 020t (s)

Qa = 00 (Nm)Qa = minus20 (Nm)Qa = minus40 (Nm)

Qa = minus60 (Nm)Qa = minus80 (Nm)Qa = minus100 (Nm)

Figure 6 Time response for a MAXON DCmotor

The driverrsquos function is to regulate the velocity whenthere are changes in the load The driver receives an inputvoltage between 0 and 5V (saturation) and transforms it to0ndash48V Additionally the driver can be configured to followa prescribed acceleration curve in order to decrease thecurrent consumed by the motor in the initial operation

The propellers used inVisor3 are theHarborModels 3540from the Wageningen B-series with four blades The coeffi-cients 119870

119879and 119870

119876can be approximated using polynomials

[37] given by

119870119879= sum

119904119905119906V119862119879

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (46)

119870119876= sum

119904119905119906V119862119876

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (47)


Table 2 Visor3rsquos thruster parameters

Parameter ValueMotor and driver

Highlow voltage driver saturation plusmn5VSlew rate 1200V sDriver gain 96

Motor gain 2248 rpmVSeawater density 1027 kgm3

PropellerDiameter 88 cmPitch 1016 cmBlade area ratio 07

Pitch ratio 11

Thrust coefficients 05

Torque coefficients 008

KT

10KQ

0

01

02

03

04

05

06

07

08

02 04 06 08 1 120Advance ratio J

Wag

enin

gen

B-se

ries p

rope

ller

Figure 7 Coefficients 119870119879and 119870

119876for Visor3rsquos propellers

Figure 7 shows 119870119879and 119870

119876curves for Visor3rsquos propellers In

order to obtain the thrust and torque coefficients a nominaladvance speed of 15ms for the vehicle was assumed Theparameters for a thruster are shown in Table 2

Visor3 has four controllable DOFs surge sway heaveand yaw Figure 8 shows the thruster location in Visor3 andTable 3 summarizes the parameters from thrusters positionand orientation in Visor3

The ROV dynamics were implemented as a continuous-time model and the EKF runs in discrete time with a 005 sfixed sample time The implementation was done using fourmain functions see Figure 9

(i) ROV Nolinear Discrete it calculates the next stateof the ROV described by (23)

Table 3 Thruster allocation

Thruster Vector position 120572 120573

1198791 [minus031 minus022 0025]m 0 01198792 [minus031 022 0025]m 0 01198793 [0 0 0]m 0 1205872

1198794 [011 0 0]m 1205872 0

011

0025

022 022

031

Y0

Z0 X0

X0

Figure 8 Thrusters position Note that all units are in meters

(ii) ROVSal Nolinear Discrete it calculates the out-put of the ROV according to the sensors

(iii) ROV linear Discrete it evaluates the Jacobian ineach state estimation described by (25)

(iv) ROVSal linear Discrete it evaluates the outputof the ROV given the Jacobian described by (28)

For the measurement matrix it was considered that Visor3has a MEMSENSE IM05-0300C050A35 triaxial micro iner-tial measurement unit (120583IMU) that provides three linearacceleration measurements and three angular velocities withnoise of 50mg and 05∘s In this work all measurements areassumed to be made in the body-fixed frame and a standardINS solution is proposed that is attitude and position areobtained through integration of signals from the IMU Themeasurement matrix is given by

H119896= [I6times6 0

6times6] (48)

To test the performance of the EKF algorithm and itsimplementation two different position references were com-manded to the ROV in the surge direction a position of 2m(see Figure 10) and in sway 1m (see Figure 10) Figure 10shows the estimation of the position in the Earth-fixed frame

A simulation experiment was conducted with a trajectorythat describes changes in the 119909 and 119910 and yaw directionsFigure 11 shows the result of the controller The experimentwas conducted with the following gain matrices

K119901= diag 10 10 10 0 0 10

K119894= diag 001 001 001 0 0 001

K119889= diag 50 50 50 0 0 50

(49)


Corrector equations

Predictor equationsInterpreted

InterpretedMATLAB Fcn

InterpretedMATLAB Fcn Interpreted

MATLAB Fcn

Matrixmultiply

Matrixmultiply

Matrixmultiply

MatrixmultiplyMatrix

multiply

Matrixmultiply

Matrixmultiply

2

2

1

1

Q

R

uOp

Divide

Inv

Product 6

Product 5

Product 4

Product 3Product 2

Transpose 1

Transpose

++

+

++

+

minus

minus

ROV_Nolinear_Discrete

ROV_linear_Discrete

ROVSal_linear_DiscreteROVSal_Nolinear_Discrete

Pk(minus)

HkPk(minus)

Pkminus1(+)

Pk(+)

++lowast

1

z

1

z

KkHkPk(minus)

kminus1(+)x

k(minus)x k(+)x

uT

uT

HTk

HkPk(minus)HTk

Hkzkzk

zk minus zkkminus1Φ

kminus1Pkminus1(+)Φkminus1Pkminus1(+)Φ ΦT

kminus1

Pk(minus)HTk

zk

Kk

ΦTkminus1

MATLABreg Fcn

Figure 9 EKF Simulink implementation

120595(t

)

MeasurementEstimation

t (s)50403020100

t (s)50403020100

t (s)50403020100

0

2

4

x(t

)

minus2

0

2

y(t

)

minus1

0

1

Figure 10 Position estimation in Earth-fixed frame

PID gains were obtained by trial and error a high pro-portional gain generates fast but aggressive response hence

a high derivative gain was added to increase damping anddecrease oscillations in spite of having a slower time responseA small integral gain was used to eliminate the steady-stateerror The PID with gravity compensation cancels the effectsof restoring forces while the vehicle is movingThis PID withcompensation has a better performance but it requires a goodestimation of the vehiclersquos attitude Additionally Figure 12shows that the regular PID is less energy-efficient due tothe high changes in the control signal This change in thecontrol signal can reduce duty life of the thrusters Finally theregular PID structure is unstable when the vehicle moves faraway from the origin due to restoring forces and propagationerror

6 Conclusions

The dynamic model of the underwater remotely operatedvehicle Visor3 has been presented This model considersforces and moments generated by the movement of thevehicle within the fluid damping and restoring forcesThe model was defined using body-fixed and Earth-fixedcoordinate systems and Visor3rsquos parameters were obtainedusing CAD models and CFD simulations

The full ROV system and the EKF were simulated tocompare the performance of the navigation system with theuse of a PID + gravity compensation control algorithm thatdepends on a good estimation of the state It is importantto state that the estimation in large periods of time candiverge due to integration of the noise present in acceleration


120595(t

)

PID

ReferencePID + compensation

8060 10020 400t (s)

0

2

4

6

8

0

5

10

15

20

25y

(t)

0

5

10

15

20

25

x(t

)

40 80 10020 600t (s)

20 40 60 80 1000t (s)

Figure 11 PID controllers with and without compensationmdashplanar motion control

measurements to obtain the velocity of the vehicle in thebody-fixed frame More sensors (such as SSBL and DVL) canbe used in Visor3 to overcome such a problem this will allowthe navigation system to get the position of the vehicle in amore precise way which is required for its regular operationsmaintenance purposes can be performed in ships hulls andunderwater structures within Colombian ports facilities

As it was shown the EKF-based navigation system iscapable of filtering the noise in measurements and accuratelyestimates the state of the vehicle this is important since anoisy signal that enters into the feedback system can causegreater efforts in the thrusters andmore energy consumptionHowever more simulations have to be carried out in orderto determine how critical is the divergence in the positionestimation caused by phenomena such as biases and lossesin the communications

Two different control algorithms were tested with thesimulation of the ROV PID and PID + gravity compensationThe PID with gravity compensation is capable of stabilizingthe system in less time compared to the PID and decreases

the energy consumption On the other hand the PIDwithoutgravity compensation loses tracking at different times thismay be caused by the force exerted from thrusters in orderto compensate oscillations from the vehicle In Visor3 theforwardbackward thrusters are nonaligned with the bodyframe therefore they can cause pitch and roll oscillationsFinally the simple PID goes unstable after some time periodHowever the PID with gravity compensation needs a goodestimation of the position and attitude This PID strategy is afirst approach to the motion control of the vehicle

Implementing the proposed navigation system and thecontroller inVisor3rsquos digital system requires the knowledge ofthe dynamic response of the vehicle and appropriate selectionof the sample time since it affects the EKF algorithmrsquos con-vergence Moreover many operations are in matrix form andwith floating-point format so the implementation of suchnavigation system and controllers requires a high computa-tion capacity of the on-board processorThese algorithms arethe first approximation to the real closed-loop control systemthat will be implemented in Visor3

International Journal of Navigation and Observation 11VT1

minus40

minus20

0

20

8060 10020 400t (s)

VT2

minus20

0

20

40

80 1000 4020 60t (s)

PIDPID + compensation

VT4

80 1000 4020 60t (s)

minus4

minus2

0

2

4


VT3

minus2

minus1

0

1

2

80 1000 4020 60t (s)

Figure 12 Control signal for each thrustermdashplanar motion control

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork was developed with the funding of FondoNacionalde Financiamiento para la Ciencia la Tecnologıa y la Inno-vacion Francisco Jose de Caldas the Colombian PetroleumCompany ECOPETROL Universidad Pontificia Bolivariana(UPB) Medellın and Universidad Nacional de ColombiaSede Medellın UNALMED through the Strategic Programfor the Development of Robotic Technology for OffshoreExploration of the Colombian Seabed Project 1210-531-30550 Contract 0265-2013

References

[1] G N Roberts ldquoTrends in marine control systemsrdquo AnnualReviews in Control vol 32 no 2 pp 263ndash269 2008

[2] M Chyba T Haberkorn R N Smith and S K ChoildquoDesign and implementation of time efficient trajectories forautonomous underwater vehiclesrdquo Ocean Engineering vol 35no 1 pp 63ndash76 2008

[3] F Xu Z-J Zou J-C Yin and J Cao ldquoIdentification modelingof underwater vehiclesrsquo nonlinear dynamics based on supportvector machinesrdquo Ocean Engineering vol 67 pp 68ndash76 2013

[4] M Candeloro E Valle M R Miyazaki R Skjetne M Lud-vigsen and A J Soslashrensen ldquoHMD as a new tool for telepresencein underwater operations and closed-loop control of ROVsrdquo inProceedings of the MTSIEEE (OCEANS rsquo15) pp 1ndash8 Washing-ton DC USA October 2015

[5] D D A Fernandes A J Soslashrensen K Y Pettersen and D CDonha ldquoOutput feedback motion control system for observa-tion class ROVs based on a high-gain state observer theoreticaland experimental resultsrdquo Control Engineering Practice vol 39pp 90ndash102 2015

[6] T I FossenGuidance and Control of Ocean Vehicles JohnWileyamp Sons New York NY USA 1994

[7] J-H Li B-H Jun P-M Lee and S-W Hong ldquoA hierarchicalreal-time control architecture for a semi-autonomous under-water vehiclerdquo Ocean Engineering vol 32 no 13 pp 1631ndash16412005

[8] H-P Tan R Diamant W K G Seah and M Waldmeyer ldquoAsurvey of techniques and challenges in underwater localizationrdquoOcean Engineering vol 38 no 14-15 pp 1663ndash1676 2011

[9] M Candeloro A J Soslashrensen S Longhi and F DukanldquoObservers for dynamic positioning of ROVswith experimentalresultsrdquo IFAC Proceedings Volumes vol 45 no 27 pp 85ndash902012 Proceedings of the 9th IFACConference onManoeuvringand Control of Marine Craft

[10] M S Grewal and A P Andrews Kalman Filtering Theory andPractice Using MATLAB JohnWiley amp Sons 2nd edition 2001

[11] M Caccia G Casalino R Cristi and G Veruggio ldquoAcousticmotion estimation and control for an unmanned underwater


vehicle in a structured environmentrdquo Control Engineering Prac-tice vol 6 no 5 pp 661ndash670 1998

[12] M Caccia and G Veruggio ldquoGuidance and control of a recon-figurable unmanned underwater vehiclerdquo Control EngineeringPractice vol 8 no 1 pp 21ndash37 2000

[13] L Drolet F Michaud and J Cote ldquoAdaptable sensor fusionusing multiple Kalman filtersrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems vol2 pp 1434ndash1439 Takamatsu Japan November 2000

[14] M Blain S Lemieux and RHoude ldquoImplementation of a ROVnavigation system using acousticDoppler sensors and Kalmanfilteringrdquo in Proceedings of the OCEANS vol 3 pp 1255ndash1260San Diego Calif USA September 2003

[15] D Loebis R Sutton J Chudley and W Naeem ldquoAdaptivetuning of a Kalman filter via fuzzy logic for an intelligent AUVnavigation systemrdquo Control Engineering Practice vol 12 no 12pp 1531ndash1539 2004

[16] J Kinsey R Eustice and LWhitcomb ldquoA survey of underwatervehicle navigation recent advances and new challengesrdquo inProceedings of the IFAC Conference of Maneuvering and Controlof Marine Craft pp 1ndash12 Lisbon Portugal 2006

[17] P-M Lee and B-H Jun ldquoPseudo long base line navigationalgorithm for underwater vehicles with inertial sensors and twoacoustic range measurementsrdquo Ocean Engineering vol 34 no3-4 pp 416ndash425 2007

[18] YWatanabe H Ochi T Shimura and T Hattori ldquoA tracking ofAUV with integration of SSBL acoustic positioning and trans-mitted INS datardquo in Proceedings of the OCEANSmdashEUROPE pp1ndash6 Bremen Germany May 2009

[19] Y Geng and J Sousa ldquoHybrid derivative-free EKF forUSBLINS tightly-coupled integration in AUVrdquo in Proceedingsof the IEEE International Conference on Autonomous andIntelligent Systems (AIS rsquo10) pp 1ndash6 IEEE Povoa de VarzimPortugal June 2010

[20] F Azis M M Aras M Rashid M Othman and S AbdullahldquoProblem identification for underwater remotely operated vehi-cle (ROV) a case studyrdquo Procedia Engineering vol 41 pp 554ndash560 2012

[21] S Cohan ldquoTrends in ROV developmentrdquo Marine TechnologySociety Journal vol 42 no 1 pp 38ndash43 2008

[22] K D Do J Pan and Z P Jiang ldquoRobust and adaptive pathfollowing for underactuated autonomous underwater vehiclesrdquoOcean Engineering vol 31 no 16 pp 1967ndash1997 2004

[23] P W J Van de Ven C Flanagan and D Toal ldquoNeural networkcontrol of underwater vehiclesrdquo Engineering Applications ofArtificial Intelligence vol 18 no 5 pp 533ndash547 2005

[24] N Q Hoang and E Kreuzer ldquoAdaptive PD-controller forpositioning of a remotely operated vehicle close to an under-water structure theory and experimentsrdquo Control EngineeringPractice vol 15 no 4 pp 411ndash419 2007

[25] W M Bessa M S Dutra and E Kreuzer ldquoDepth control ofremotely operated underwater vehicles using an adaptive fuzzyslidingmode controllerrdquoRobotics andAutonomous Systems vol56 no 8 pp 670ndash677 2008

[26] A Alvarez A Caffaz A Caiti et al ldquoFolaga a low-costautonomous underwater vehicle combining glider and AUVcapabilitiesrdquo Ocean Engineering vol 36 no 1 pp 24ndash38 2009

[27] B Subudhi K Mukherjee and S Ghosh ldquoA static outputfeedback control design for path following of autonomousunderwater vehicle in vertical planerdquo Ocean Engineering vol63 pp 72ndash76 2013

[28] K Ishaque S S Abdullah S M Ayob and Z Salam ldquoAsimplified approach to design fuzzy logic controller for anunderwater vehiclerdquo Ocean Engineering vol 38 no 1 pp 271ndash284 2011

[29] P Herman ldquoDecoupled PD set-point controller for underwatervehiclesrdquo Ocean Engineering vol 36 no 7 pp 529ndash534 2009

[30] J Petrich and D J Stilwell ldquoRobust control for an autonomousunderwater vehicle that suppresses pitch and yaw couplingrdquoOcean Engineering vol 38 no 1 pp 197ndash204 2011

[31] L B Gutierrez C A Zuluaga J A Ramırez et al ldquoDevelop-ment of an underwater remotely operated vehicle (ROV) forsurveillance and inspection of port facilitiesrdquo in Proceedings ofthe ASME 2010 International Mechanical Engineering Congressand Exposition pp 631ndash640 Vancouver Canada 2010

[32] L M Aristizabal S Rua C E Gaviria et al ldquoDesign of anopen source-based control platform for an underwater remotelyoperated vehiclerdquo DYNA vol 83 no 195 pp 198ndash205 2016

[33] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons London UK 2011

[34] F DukanROVmotion control systems [PhD thesis] NorwegianUniversity of Science and Technology (NTNU) TrondheimNorway 2014

[35] Maxon EC Motor ldquoEC 45 ⊘45mm brushless 150 Wattrdquo 2015[36] Maxon-Motor Planetary Gearhead GP 42 C 042mm 3ndash15Nm

2015[37] M Bernitsas D Ray and P Kinley ldquoKT KQ and efficiency

curves for the Wageningen B-series propellersrdquo Tech Rep 237Department of Naval Architecture and Marine EngineeringCollege of Engineering The University of Michigan AnnArbor Mich USA 1981

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



High level

Mid-level

Low level

Mission

Mission planning

Waypoints

Path planning

Setpoints

Control Vehicle

Position

Sensors

Navigationsystem

velocityattitude

Figure 1 Control structure for an underwater vehicle

the vehicle Caccia et al [11] developed a Kalman-filter-based acoustic navigationmodule to control theUUVRoby2Caccia and Veruggio [12] used Kalman-filter techniques toestimate the state using different sampling rates of a depth-meter and altimeter Drolet et al [13] presented an integratedsensor fusion strategy using multiple Kalman filters allowingdifferent combination of sensors Blain et al [14] developedand tested a Kalman filter to merge data from an acousticpositioning system a bathymeter and a DVL Loebis et al[15] implemented an intelligent navigation system based onthe integrated use of the global positioning system (GPS)and several inertial navigation system (INS) sensors forAUV Kinsey et al [16] presented a survey with advances inunderwater vehicle navigation and identified future researchchallenges Lee and Jun [17] presented a pseudo long baseline(LBL) navigation algorithm using the EKF Watanabe et al[18] proposed an accurate tracking method to estimate AUVposition by using a super-short baseline (SSBL) from theship Geng and Sousa [19] presented a hybrid derivative-free extended Kalman filter taking advantage of both thelinear time propagation of the Kalman filter and nonlinearmeasurement propagation of the derivative-free extendedKalman filter

The control system is another component of the lowerlevel of the control structure It contains a set of algorithmsthat stabilize the state of the vehicle so it can follow thecommands generated in the path planning system Thecontrol of an underwater vehicle is complex because thereare highly nonlinear hydrodynamic effects resulting from theinteraction with the environment that cannot be quantified[20] Cohan [21] states that the development of controlsystems for ROVs is a current and promising topic for futuredevelopments this can be verified with the number of papersthat can be found in literature

Caccia and Veruggio [12] implemented and tested aguidance and control system for underwater vehicles usingprogrammed controllers to regulate speed at the low level ina hierarchical three-level structure Do et al [22] developeda robust adaptive control strategy to ensure that a six-degree-of-freedom vehicle follows a prescribed path usingfour actuators Van de Ven et al [23] presented a qualitativeassessment of the performance of control strategies usingneural networks indicating the advantages disadvantagesand application recommendations Hoang and Kreuzer [24]

designed an adaptive PD controller for dynamic positioningof ROVs when the mission is executed in places nearsubmerged structures and requires great execution precisionBessa et al [25] used slidingmode controllers combinedwithfuzzy adaptive algorithms for controlling depth in ROVsAlvarez et al [26] developed a robust PID controller forcontrolling AUV used in oceanographic sampling workSubudhi et al [27] presented the design of a feedbackcontroller for tracking paths in vertical planes Ishaque etal [28] presented a simplification of the conventional fuzzycontroller for an underwater vehicle Herman [29] presenteda decoupled PD setpoint controller which is expressed interms of quasi-velocities for underwater vehicles Petrich andStilwell [30] presented a robust control for an autonomousunderwater vehicle that suppresses pitch and yaw coupling

Gutierrez et al [31] developed the underwater remotelyoperated vehicle Visor3 for surveillance and maintenanceof ship hulls and underwater structures of Colombian portfacilities and oceanographic research Visor3 has a three-layerhardware architecture instrumentation communicationsand control [32] Although much work has been done inmechanics and electronics a closed-loop control system wasnot developed for the ROV Visor3 so the capabilities of thevehicle are still completely dependent on the pilot skills

This work addresses the first approach to develop thelow-level control system for the ROV Visor3 that will helpoperators to maneuver and determine the position andattitude of the vehicle under certain scenarios such as in portsor dams inspection tasks The second section presents themathematical model of the vehicle the third section showsthe development of the nonlinear model-based observerand the fourth section presents the implemented controlalgorithms then the simulation results are shown and someconclusions are presented

2 Mathematical Model

To analyze themotion ofVisor3 in a three-dimensional spacetwo coordinate frames are defined an inertial Earth-fixedframe (this reference frame is considered as the North-East-Down (NED) frame attached to a port facility) where themotion of the vehicle is described and a body-fixed framewhich is conveniently fixed to the vehicle and moves withit Figure 2 The acceleration of the Earth due to rotation isneglected for this work The position and orientation of thevehicle are described relative to the Earth-fixed frame as

120578 = [119909 119910 119911 120601 120579 120595]T (1)

where 1205781= [119909 119910 119911]

T is the position and 1205782= [120601 120579 120595]

T

is the orientation The linear and angular velocities of thevehicle relative to the body-fixed frame are given by

^ = [119906 V 119908 119901 119902 119903]T (2)

where ^1= [119906 V 119908]T and ^

2= [119901 119902 119903]

T are the linear andangular velocities respectively


Y0 Y

X0

Incoming-flow Z0

Body-fixed

r0

X

Z

Earth-fixed

Xf

minus120573

120572

(sway)q (pitch)

p (roll)

u (surge)

r (yaw)

w (heave)



M ^ + C (^) ^ +D (^) ^ + g (120578) = 120591

= J (120578) ^(3)







MRBk + CRB (k) k = 120591RB (4)



minus119898S (r119866)

119898S (r119866) I

0

] (5)





CRB = [CRB11

CRB12

CRB21

CRB22

] (6)

whereCRB11

= 03times3

CRB12

= minus119898S (^1) minus 119898S (^

2) S (r119866)

CRB21

= minus119898S (^1) + 119898S (^

2) S (r119866)

CRB22

= minusS (I0) ^2

(7)


M119860= [

A11

3times3 A12

3times3

A21

3times3 A22

3times3]

= minus[

120597120591

120597

120597120591

120597V120597120591

120597

120597120591

120597

120597120591

120597

120597120591

120597 119903

]

(8)




M119860= minus diag 119883

119884V 119885 119870119872 119873 119903 (9)


C119860(^) = [

03times3

minusS (A11^1+ A12^2)


21^1+ A22^2)

] (10)


D (^) = minus diag 119883119906 119884V 119885119908 119870119901119872119902 119873119903

minus diag 119883119906|119906| |119906| 119884V|V| |V| 119885119908|119908| |119908| 119870119901|119901|

10038161003816100381610038161199011003816100381610038161003816

119872119902|119902|

10038161003816100381610038161199021003816100381610038161003816 119873119903|119903| |119903|

(11)


g (120578) =

[

[

[

[

[

[

[

[

[

[

[

[


119910119887119861 cos 120579 cos120601 minus 119911

119887119861 cos 120579 sin120601

minus119911119887119861 sin 120579 minus 119909

119887119861 cos 120579 cos120601

119909119887119861 cos 120579 sin120601 + 119910

119887119861 sin 120579

]

]

]

]

]

]

]

]

]

]

]

]

(12)




J1(1205782)

=[

[

[


]

]

]

(13)


J2(1205782) =

[

[

[

[

[

1 s120601t120579 c120601t1205790 c120601 minuss120601

0

s120601c120579

c120601c120579

]

]

]

]

]

(14)


expressed as

[

1

2

] = [

J1(1205782) 03times3

03times3

J2(1205782)

] [

^1

^2

] (15)



x = 119891 (x 119905) + Bk (119905) +m (119905) (16)

z = ℎ (x 119905) + n (119905) (17)



J (120578) ^] (18)


B = [Mminus1

06times6

] (19)



119864 ⟨m (119905)⟩ = 0

119864 ⟨m (119905)mT(119904)⟩ = 120575 (119905 minus 119904)Q (119905)

119864 ⟨n (119905)⟩ = 0

119864 ⟨n (119905)nT(119904)⟩ = 120575 (119905 minus 119904)R (119905)

(20)


x119896= 119892119896minus1(x119896minus1) + Δk


z119896= ℎ119896(x119896) + n119896

(21)

where


119896minus1 119905119896minus1)

Δ = ℎB(22)


^119896= ^119896minus1+ ℎMminus1 [120591


119896minus1) ^119896minus1


119896minus1)]

120578119896= 120578119896minus1+ ℎ [J (120578

119896minus1) ^119896]

(23)


x119896(minus) = 119892

119896minus1(x119896minus1(+)) + Δk

119896minus1

z119896= ℎ119896(x119896(minus))

P119896(minus) = Φ



(24)


119896(+) is the a



119896(+) is the a



Φ119896minus1asymp

120597119892119896

120597x

10038161003816100381610038161003816100381610038161003816x=x119896minus1(minus)

(25)


x119896(+) = x

119896(minus) + K

119896(z119896minus z119896)

P119896(+) = [I minus K

119896H119896]P119896(minus)

(26)


K119896= P119896(minus)HT

119896[H119896P119896(minus)HT

119896+ R119896]

minus1

(27)




Pilot Joystick B+

minus minus

usum

g

C + D


120578120591 ^



H119896asymp

120597ℎ119896

120597x

10038161003816100381610038161003816100381610038161003816x=x119896(minus)

(28)


119896minus1and H









f = [1198911 1198912 sdot sdot sdot 119891119903]T (29)


120591 = Tf (30)






computed as

t119894= [

I3times3

minusST (r119887119905119894119887)

]R (120572 120573)[[[

1

0

0

]

]

]

119891119894 (32)


R (120572 120573) = R119911120572R119910120573 (33)

where R119911120572

and R119910120573

are described by

R119911120572=[

[

[

c120572 minuss120572 0s120572 c120572 00 0 1

]

]

]

(34)

R119910120573=[

[

[

c120573 0 s1205730 1 0

minuss120573 0 c120573

]

]

]

(35)




120591 = TKu (36)


119870119894= 119870119879(119869) 120588119863

4 (37)





Tdagger = TT(TTT

)

minus1

(39)




V119894= (

1

119866119898119894

)(

1

119866119889119894

) sgn (119906119894)radic1003816100381610038161003816119906119894

1003816100381610038161003816 (41)

where 119866119898119894

and 119866119889119894


119894


119872119892= (1119866

119898119894

)(1119866119889119894

)


120591PID = K119901e (119905) + K119889e (119905) + K119894 int119905

0

e (120591) 119889120591 (42)




e = 120578119889minus 120578 (43)


119901 K119894 and K



i(V

)

5 10 15 200ui (rps)

Mg = 01

Mg = 03

Mg = 05

Mg = 07

Mg = 09

Mg = 11

0

1

2

3

4

5


allocation problem


120591 = JT (120578) 120591PID + g (120578) (44)










Sensors


B+

minus

+


uJint int

C + DJT(middot)

g(middot)

g(middot)

Tdagger120578p 120578d 120578

times

^




29 kgm2

119868119910119910

25 kgm2

119868119911119911

30 kgm2

119868119909119910


119868119909119911


119868119910119911



[119909119892 119910119892 119911119892] [0 0 0]m

[119909119887 119910119887 119911119887] [17 18 68] times 10

minus3m119883

65 kg119884V 598 kg119885

598 kg119870

0 kgm2

119872

22 kgm2

119873119903

22 kgm2

119883119906|119906|


minus1008 kgm119870119901|119901|

minus4003 kgm2

119872119902|119902|

minus1008 kgm2

119873119903|119903|

minus1008 kgm2



119866119898=

Velmax119881nom

(45)


119898

0

200

400

600

800

1000

1200

120596(t)

(rpm

)

005 01 015 020t (s)






119879and 119870


[37] given by

119870119879= sum

119904119905119906V119862119879

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (46)

119870119876= sum

119904119905119906V119862119876

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (47)







Pitch ratio 11



KT

10KQ

0

01

02

03

04

05

06

07

08


Wag

enin

gen

B-se

ries p

rope

ller



Figure 7 shows 119870119879and 119870









1198794 [011 0 0]m 1205872 0

011

0025

022 022

031

Y0

Z0 X0

X0






H119896= [I6times6 0

6times6] (48)



K119901= diag 10 10 10 0 0 10

K119894= diag 001 001 001 0 0 001

K119889= diag 50 50 50 0 0 50

(49)


Corrector equations




MATLAB Fcn

Matrixmultiply

Matrixmultiply

Matrixmultiply


multiply

Matrixmultiply

Matrixmultiply

2

2

1

1

Q

R

uOp

Divide

Inv

Product 6

Product 5

Product 4

Product 3Product 2

Transpose 1

Transpose

++

+

++

+

minus

minus


ROV_linear_Discrete


Pk(minus)

HkPk(minus)

Pkminus1(+)

Pk(+)

++lowast

1

z

1

z

KkHkPk(minus)

kminus1(+)x

k(minus)x k(+)x

uT

uT

HTk

HkPk(minus)HTk

Hkzkzk



kminus1

Pk(minus)HTk

zk

Kk

ΦTkminus1

MATLABreg Fcn


120595(t

)


t (s)50403020100

t (s)50403020100

t (s)50403020100

0

2

4

x(t

)

minus2

0

2

y(t

)

minus1

0

1




6 Conclusions




120595(t

)

PID


8060 10020 400t (s)

0

2

4

6

8

0

5

10

15

20

25y

(t)

0

5

10

15

20

25

x(t

)

40 80 10020 600t (s)

20 40 60 80 1000t (s)








minus40

minus20

0

20

8060 10020 400t (s)

VT2

minus20

0

20

40

80 1000 4020 60t (s)


VT4

80 1000 4020 60t (s)

minus4

minus2

0

2

4


VT3

minus2

minus1

0

1

2

80 1000 4020 60t (s)


Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Y0 Y

X0

Incoming-flow Z0

Body-fixed

r0

X

Z

Earth-fixed

Xf

minus120573

120572

(sway)q (pitch)

p (roll)

u (surge)

r (yaw)

w (heave)



M ^ + C (^) ^ +D (^) ^ + g (120578) = 120591

= J (120578) ^(3)







MRBk + CRB (k) k = 120591RB (4)



minus119898S (r119866)

119898S (r119866) I

0

] (5)





CRB = [CRB11

CRB12

CRB21

CRB22

] (6)

whereCRB11

= 03times3

CRB12

= minus119898S (^1) minus 119898S (^

2) S (r119866)

CRB21

= minus119898S (^1) + 119898S (^

2) S (r119866)

CRB22

= minusS (I0) ^2

(7)


M119860= [

A11

3times3 A12

3times3

A21

3times3 A22

3times3]

= minus[

120597120591

120597

120597120591

120597V120597120591

120597

120597120591

120597

120597120591

120597

120597120591

120597 119903

]

(8)




M119860= minus diag 119883

119884V 119885 119870119872 119873 119903 (9)


C119860(^) = [

03times3

minusS (A11^1+ A12^2)


21^1+ A22^2)

] (10)


D (^) = minus diag 119883119906 119884V 119885119908 119870119901119872119902 119873119903

minus diag 119883119906|119906| |119906| 119884V|V| |V| 119885119908|119908| |119908| 119870119901|119901|

10038161003816100381610038161199011003816100381610038161003816

119872119902|119902|

10038161003816100381610038161199021003816100381610038161003816 119873119903|119903| |119903|

(11)


g (120578) =

[

[

[

[

[

[

[

[

[

[

[

[


119910119887119861 cos 120579 cos120601 minus 119911

119887119861 cos 120579 sin120601

minus119911119887119861 sin 120579 minus 119909

119887119861 cos 120579 cos120601

119909119887119861 cos 120579 sin120601 + 119910

119887119861 sin 120579

]

]

]

]

]

]

]

]

]

]

]

]

(12)




J1(1205782)

=[

[

[


]

]

]

(13)


J2(1205782) =

[

[

[

[

[

1 s120601t120579 c120601t1205790 c120601 minuss120601

0

s120601c120579

c120601c120579

]

]

]

]

]

(14)


expressed as

[

1

2

] = [

J1(1205782) 03times3

03times3

J2(1205782)

] [

^1

^2

] (15)



x = 119891 (x 119905) + Bk (119905) +m (119905) (16)

z = ℎ (x 119905) + n (119905) (17)



J (120578) ^] (18)


B = [Mminus1

06times6

] (19)



119864 ⟨m (119905)⟩ = 0

119864 ⟨m (119905)mT(119904)⟩ = 120575 (119905 minus 119904)Q (119905)

119864 ⟨n (119905)⟩ = 0

119864 ⟨n (119905)nT(119904)⟩ = 120575 (119905 minus 119904)R (119905)

(20)


x119896= 119892119896minus1(x119896minus1) + Δk


z119896= ℎ119896(x119896) + n119896

(21)

where


119896minus1 119905119896minus1)

Δ = ℎB(22)


^119896= ^119896minus1+ ℎMminus1 [120591


119896minus1) ^119896minus1


119896minus1)]

120578119896= 120578119896minus1+ ℎ [J (120578

119896minus1) ^119896]

(23)


x119896(minus) = 119892

119896minus1(x119896minus1(+)) + Δk

119896minus1

z119896= ℎ119896(x119896(minus))

P119896(minus) = Φ



(24)


119896(+) is the a



119896(+) is the a



Φ119896minus1asymp

120597119892119896

120597x

10038161003816100381610038161003816100381610038161003816x=x119896minus1(minus)

(25)


x119896(+) = x

119896(minus) + K

119896(z119896minus z119896)

P119896(+) = [I minus K

119896H119896]P119896(minus)

(26)


K119896= P119896(minus)HT

119896[H119896P119896(minus)HT

119896+ R119896]

minus1

(27)




Pilot Joystick B+

minus minus

usum

g

C + D


120578120591 ^



H119896asymp

120597ℎ119896

120597x

10038161003816100381610038161003816100381610038161003816x=x119896(minus)

(28)


119896minus1and H









f = [1198911 1198912 sdot sdot sdot 119891119903]T (29)


120591 = Tf (30)






computed as

t119894= [

I3times3

minusST (r119887119905119894119887)

]R (120572 120573)[[[

1

0

0

]

]

]

119891119894 (32)


R (120572 120573) = R119911120572R119910120573 (33)

where R119911120572

and R119910120573

are described by

R119911120572=[

[

[

c120572 minuss120572 0s120572 c120572 00 0 1

]

]

]

(34)

R119910120573=[

[

[

c120573 0 s1205730 1 0

minuss120573 0 c120573

]

]

]

(35)




120591 = TKu (36)


119870119894= 119870119879(119869) 120588119863

4 (37)





Tdagger = TT(TTT

)

minus1

(39)




V119894= (

1

119866119898119894

)(

1

119866119889119894

) sgn (119906119894)radic1003816100381610038161003816119906119894

1003816100381610038161003816 (41)

where 119866119898119894

and 119866119889119894


119894


119872119892= (1119866

119898119894

)(1119866119889119894

)


120591PID = K119901e (119905) + K119889e (119905) + K119894 int119905

0

e (120591) 119889120591 (42)




e = 120578119889minus 120578 (43)


119901 K119894 and K



i(V

)

5 10 15 200ui (rps)

Mg = 01

Mg = 03

Mg = 05

Mg = 07

Mg = 09

Mg = 11

0

1

2

3

4

5


allocation problem


120591 = JT (120578) 120591PID + g (120578) (44)










Sensors


B+

minus

+


uJint int

C + DJT(middot)

g(middot)

g(middot)

Tdagger120578p 120578d 120578

times

^




29 kgm2

119868119910119910

25 kgm2

119868119911119911

30 kgm2

119868119909119910


119868119909119911


119868119910119911



[119909119892 119910119892 119911119892] [0 0 0]m

[119909119887 119910119887 119911119887] [17 18 68] times 10

minus3m119883

65 kg119884V 598 kg119885

598 kg119870

0 kgm2

119872

22 kgm2

119873119903

22 kgm2

119883119906|119906|


minus1008 kgm119870119901|119901|

minus4003 kgm2

119872119902|119902|

minus1008 kgm2

119873119903|119903|

minus1008 kgm2



119866119898=

Velmax119881nom

(45)


119898

0

200

400

600

800

1000

1200

120596(t)

(rpm

)

005 01 015 020t (s)






119879and 119870


[37] given by

119870119879= sum

119904119905119906V119862119879

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (46)

119870119876= sum

119904119905119906V119862119876

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (47)







Pitch ratio 11



KT

10KQ

0

01

02

03

04

05

06

07

08


Wag

enin

gen

B-se

ries p

rope

ller



Figure 7 shows 119870119879and 119870









1198794 [011 0 0]m 1205872 0

011

0025

022 022

031

Y0

Z0 X0

X0






H119896= [I6times6 0

6times6] (48)



K119901= diag 10 10 10 0 0 10

K119894= diag 001 001 001 0 0 001

K119889= diag 50 50 50 0 0 50

(49)


Corrector equations




MATLAB Fcn

Matrixmultiply

Matrixmultiply

Matrixmultiply


multiply

Matrixmultiply

Matrixmultiply

2

2

1

1

Q

R

uOp

Divide

Inv

Product 6

Product 5

Product 4

Product 3Product 2

Transpose 1

Transpose

++

+

++

+

minus

minus


ROV_linear_Discrete


Pk(minus)

HkPk(minus)

Pkminus1(+)

Pk(+)

++lowast

1

z

1

z

KkHkPk(minus)

kminus1(+)x

k(minus)x k(+)x

uT

uT

HTk

HkPk(minus)HTk

Hkzkzk



kminus1

Pk(minus)HTk

zk

Kk

ΦTkminus1

MATLABreg Fcn


120595(t

)


t (s)50403020100

t (s)50403020100

t (s)50403020100

0

2

4

x(t

)

minus2

0

2

y(t

)

minus1

0

1




6 Conclusions




120595(t

)

PID


8060 10020 400t (s)

0

2

4

6

8

0

5

10

15

20

25y

(t)

0

5

10

15

20

25

x(t

)

40 80 10020 600t (s)

20 40 60 80 1000t (s)








minus40

minus20

0

20

8060 10020 400t (s)

VT2

minus20

0

20

40

80 1000 4020 60t (s)


VT4

80 1000 4020 60t (s)

minus4

minus2

0

2

4


VT3

minus2

minus1

0

1

2

80 1000 4020 60t (s)


Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











J1(1205782)

=[

[

[


]

]

]

(13)


J2(1205782) =

[

[

[

[

[

1 s120601t120579 c120601t1205790 c120601 minuss120601

0

s120601c120579

c120601c120579

]

]

]

]

]

(14)


expressed as

[

1

2

] = [

J1(1205782) 03times3

03times3

J2(1205782)

] [

^1

^2

] (15)



x = 119891 (x 119905) + Bk (119905) +m (119905) (16)

z = ℎ (x 119905) + n (119905) (17)



J (120578) ^] (18)


B = [Mminus1

06times6

] (19)



119864 ⟨m (119905)⟩ = 0

119864 ⟨m (119905)mT(119904)⟩ = 120575 (119905 minus 119904)Q (119905)

119864 ⟨n (119905)⟩ = 0

119864 ⟨n (119905)nT(119904)⟩ = 120575 (119905 minus 119904)R (119905)

(20)


x119896= 119892119896minus1(x119896minus1) + Δk


z119896= ℎ119896(x119896) + n119896

(21)

where


119896minus1 119905119896minus1)

Δ = ℎB(22)


^119896= ^119896minus1+ ℎMminus1 [120591


119896minus1) ^119896minus1


119896minus1)]

120578119896= 120578119896minus1+ ℎ [J (120578

119896minus1) ^119896]

(23)


x119896(minus) = 119892

119896minus1(x119896minus1(+)) + Δk

119896minus1

z119896= ℎ119896(x119896(minus))

P119896(minus) = Φ



(24)


119896(+) is the a



119896(+) is the a



Φ119896minus1asymp

120597119892119896

120597x

10038161003816100381610038161003816100381610038161003816x=x119896minus1(minus)

(25)


x119896(+) = x

119896(minus) + K

119896(z119896minus z119896)

P119896(+) = [I minus K

119896H119896]P119896(minus)

(26)


K119896= P119896(minus)HT

119896[H119896P119896(minus)HT

119896+ R119896]

minus1

(27)




Pilot Joystick B+

minus minus

usum

g

C + D


120578120591 ^



H119896asymp

120597ℎ119896

120597x

10038161003816100381610038161003816100381610038161003816x=x119896(minus)

(28)


119896minus1and H









f = [1198911 1198912 sdot sdot sdot 119891119903]T (29)


120591 = Tf (30)






computed as

t119894= [

I3times3

minusST (r119887119905119894119887)

]R (120572 120573)[[[

1

0

0

]

]

]

119891119894 (32)


R (120572 120573) = R119911120572R119910120573 (33)

where R119911120572

and R119910120573

are described by

R119911120572=[

[

[

c120572 minuss120572 0s120572 c120572 00 0 1

]

]

]

(34)

R119910120573=[

[

[

c120573 0 s1205730 1 0

minuss120573 0 c120573

]

]

]

(35)




120591 = TKu (36)


119870119894= 119870119879(119869) 120588119863

4 (37)





Tdagger = TT(TTT

)

minus1

(39)




V119894= (

1

119866119898119894

)(

1

119866119889119894

) sgn (119906119894)radic1003816100381610038161003816119906119894

1003816100381610038161003816 (41)

where 119866119898119894

and 119866119889119894


119894


119872119892= (1119866

119898119894

)(1119866119889119894

)


120591PID = K119901e (119905) + K119889e (119905) + K119894 int119905

0

e (120591) 119889120591 (42)




e = 120578119889minus 120578 (43)


119901 K119894 and K



i(V

)

5 10 15 200ui (rps)

Mg = 01

Mg = 03

Mg = 05

Mg = 07

Mg = 09

Mg = 11

0

1

2

3

4

5


allocation problem


120591 = JT (120578) 120591PID + g (120578) (44)










Sensors


B+

minus

+


uJint int

C + DJT(middot)

g(middot)

g(middot)

Tdagger120578p 120578d 120578

times

^




29 kgm2

119868119910119910

25 kgm2

119868119911119911

30 kgm2

119868119909119910


119868119909119911


119868119910119911



[119909119892 119910119892 119911119892] [0 0 0]m

[119909119887 119910119887 119911119887] [17 18 68] times 10

minus3m119883

65 kg119884V 598 kg119885

598 kg119870

0 kgm2

119872

22 kgm2

119873119903

22 kgm2

119883119906|119906|


minus1008 kgm119870119901|119901|

minus4003 kgm2

119872119902|119902|

minus1008 kgm2

119873119903|119903|

minus1008 kgm2



119866119898=

Velmax119881nom

(45)


119898

0

200

400

600

800

1000

1200

120596(t)

(rpm

)

005 01 015 020t (s)






119879and 119870


[37] given by

119870119879= sum

119904119905119906V119862119879

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (46)

119870119876= sum

119904119905119906V119862119876

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (47)







Pitch ratio 11



KT

10KQ

0

01

02

03

04

05

06

07

08


Wag

enin

gen

B-se

ries p

rope

ller



Figure 7 shows 119870119879and 119870









1198794 [011 0 0]m 1205872 0

011

0025

022 022

031

Y0

Z0 X0

X0






H119896= [I6times6 0

6times6] (48)



K119901= diag 10 10 10 0 0 10

K119894= diag 001 001 001 0 0 001

K119889= diag 50 50 50 0 0 50

(49)


Corrector equations




MATLAB Fcn

Matrixmultiply

Matrixmultiply

Matrixmultiply


multiply

Matrixmultiply

Matrixmultiply

2

2

1

1

Q

R

uOp

Divide

Inv

Product 6

Product 5

Product 4

Product 3Product 2

Transpose 1

Transpose

++

+

++

+

minus

minus


ROV_linear_Discrete


Pk(minus)

HkPk(minus)

Pkminus1(+)

Pk(+)

++lowast

1

z

1

z

KkHkPk(minus)

kminus1(+)x

k(minus)x k(+)x

uT

uT

HTk

HkPk(minus)HTk

Hkzkzk



kminus1

Pk(minus)HTk

zk

Kk

ΦTkminus1

MATLABreg Fcn


120595(t

)


t (s)50403020100

t (s)50403020100

t (s)50403020100

0

2

4

x(t

)

minus2

0

2

y(t

)

minus1

0

1




6 Conclusions




120595(t

)

PID


8060 10020 400t (s)

0

2

4

6

8

0

5

10

15

20

25y

(t)

0

5

10

15

20

25

x(t

)

40 80 10020 600t (s)

20 40 60 80 1000t (s)








minus40

minus20

0

20

8060 10020 400t (s)

VT2

minus20

0

20

40

80 1000 4020 60t (s)


VT4

80 1000 4020 60t (s)

minus4

minus2

0

2

4


VT3

minus2

minus1

0

1

2

80 1000 4020 60t (s)


Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Pilot Joystick B+

minus minus

usum

g

C + D


120578120591 ^



H119896asymp

120597ℎ119896

120597x

10038161003816100381610038161003816100381610038161003816x=x119896(minus)

(28)


119896minus1and H









f = [1198911 1198912 sdot sdot sdot 119891119903]T (29)


120591 = Tf (30)






computed as

t119894= [

I3times3

minusST (r119887119905119894119887)

]R (120572 120573)[[[

1

0

0

]

]

]

119891119894 (32)


R (120572 120573) = R119911120572R119910120573 (33)

where R119911120572

and R119910120573

are described by

R119911120572=[

[

[

c120572 minuss120572 0s120572 c120572 00 0 1

]

]

]

(34)

R119910120573=[

[

[

c120573 0 s1205730 1 0

minuss120573 0 c120573

]

]

]

(35)




120591 = TKu (36)


119870119894= 119870119879(119869) 120588119863

4 (37)





Tdagger = TT(TTT

)

minus1

(39)




V119894= (

1

119866119898119894

)(

1

119866119889119894

) sgn (119906119894)radic1003816100381610038161003816119906119894

1003816100381610038161003816 (41)

where 119866119898119894

and 119866119889119894


119894


119872119892= (1119866

119898119894

)(1119866119889119894

)


120591PID = K119901e (119905) + K119889e (119905) + K119894 int119905

0

e (120591) 119889120591 (42)




e = 120578119889minus 120578 (43)


119901 K119894 and K



i(V

)

5 10 15 200ui (rps)

Mg = 01

Mg = 03

Mg = 05

Mg = 07

Mg = 09

Mg = 11

0

1

2

3

4

5


allocation problem


120591 = JT (120578) 120591PID + g (120578) (44)










Sensors


B+

minus

+


uJint int

C + DJT(middot)

g(middot)

g(middot)

Tdagger120578p 120578d 120578

times

^




29 kgm2

119868119910119910

25 kgm2

119868119911119911

30 kgm2

119868119909119910


119868119909119911


119868119910119911



[119909119892 119910119892 119911119892] [0 0 0]m

[119909119887 119910119887 119911119887] [17 18 68] times 10

minus3m119883

65 kg119884V 598 kg119885

598 kg119870

0 kgm2

119872

22 kgm2

119873119903

22 kgm2

119883119906|119906|


minus1008 kgm119870119901|119901|

minus4003 kgm2

119872119902|119902|

minus1008 kgm2

119873119903|119903|

minus1008 kgm2



119866119898=

Velmax119881nom

(45)


119898

0

200

400

600

800

1000

1200

120596(t)

(rpm

)

005 01 015 020t (s)






119879and 119870


[37] given by

119870119879= sum

119904119905119906V119862119879

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (46)

119870119876= sum

119904119905119906V119862119876

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (47)







Pitch ratio 11



KT

10KQ

0

01

02

03

04

05

06

07

08


Wag

enin

gen

B-se

ries p

rope

ller



Figure 7 shows 119870119879and 119870









1198794 [011 0 0]m 1205872 0

011

0025

022 022

031

Y0

Z0 X0

X0






H119896= [I6times6 0

6times6] (48)



K119901= diag 10 10 10 0 0 10

K119894= diag 001 001 001 0 0 001

K119889= diag 50 50 50 0 0 50

(49)


Corrector equations




MATLAB Fcn

Matrixmultiply

Matrixmultiply

Matrixmultiply


multiply

Matrixmultiply

Matrixmultiply

2

2

1

1

Q

R

uOp

Divide

Inv

Product 6

Product 5

Product 4

Product 3Product 2

Transpose 1

Transpose

++

+

++

+

minus

minus


ROV_linear_Discrete


Pk(minus)

HkPk(minus)

Pkminus1(+)

Pk(+)

++lowast

1

z

1

z

KkHkPk(minus)

kminus1(+)x

k(minus)x k(+)x

uT

uT

HTk

HkPk(minus)HTk

Hkzkzk



kminus1

Pk(minus)HTk

zk

Kk

ΦTkminus1

MATLABreg Fcn


120595(t

)


t (s)50403020100

t (s)50403020100

t (s)50403020100

0

2

4

x(t

)

minus2

0

2

y(t

)

minus1

0

1




6 Conclusions




120595(t

)

PID


8060 10020 400t (s)

0

2

4

6

8

0

5

10

15

20

25y

(t)

0

5

10

15

20

25

x(t

)

40 80 10020 600t (s)

20 40 60 80 1000t (s)








minus40

minus20

0

20

8060 10020 400t (s)

VT2

minus20

0

20

40

80 1000 4020 60t (s)


VT4

80 1000 4020 60t (s)

minus4

minus2

0

2

4


VT3

minus2

minus1

0

1

2

80 1000 4020 60t (s)


Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120591 = TKu (36)


119870119894= 119870119879(119869) 120588119863

4 (37)





Tdagger = TT(TTT

)

minus1

(39)




V119894= (

1

119866119898119894

)(

1

119866119889119894

) sgn (119906119894)radic1003816100381610038161003816119906119894

1003816100381610038161003816 (41)

where 119866119898119894

and 119866119889119894


119894


119872119892= (1119866

119898119894

)(1119866119889119894

)


120591PID = K119901e (119905) + K119889e (119905) + K119894 int119905

0

e (120591) 119889120591 (42)




e = 120578119889minus 120578 (43)


119901 K119894 and K



i(V

)

5 10 15 200ui (rps)

Mg = 01

Mg = 03

Mg = 05

Mg = 07

Mg = 09

Mg = 11

0

1

2

3

4

5


allocation problem


120591 = JT (120578) 120591PID + g (120578) (44)










Sensors


B+

minus

+


uJint int

C + DJT(middot)

g(middot)

g(middot)

Tdagger120578p 120578d 120578

times

^




29 kgm2

119868119910119910

25 kgm2

119868119911119911

30 kgm2

119868119909119910


119868119909119911


119868119910119911



[119909119892 119910119892 119911119892] [0 0 0]m

[119909119887 119910119887 119911119887] [17 18 68] times 10

minus3m119883

65 kg119884V 598 kg119885

598 kg119870

0 kgm2

119872

22 kgm2

119873119903

22 kgm2

119883119906|119906|


minus1008 kgm119870119901|119901|

minus4003 kgm2

119872119902|119902|

minus1008 kgm2

119873119903|119903|

minus1008 kgm2



119866119898=

Velmax119881nom

(45)


119898

0

200

400

600

800

1000

1200

120596(t)

(rpm

)

005 01 015 020t (s)






119879and 119870


[37] given by

119870119879= sum

119904119905119906V119862119879

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (46)

119870119876= sum

119904119905119906V119862119876

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (47)







Pitch ratio 11



KT

10KQ

0

01

02

03

04

05

06

07

08


Wag

enin

gen

B-se

ries p

rope

ller



Figure 7 shows 119870119879and 119870









1198794 [011 0 0]m 1205872 0

011

0025

022 022

031

Y0

Z0 X0

X0






H119896= [I6times6 0

6times6] (48)



K119901= diag 10 10 10 0 0 10

K119894= diag 001 001 001 0 0 001

K119889= diag 50 50 50 0 0 50

(49)


Corrector equations




MATLAB Fcn

Matrixmultiply

Matrixmultiply

Matrixmultiply


multiply

Matrixmultiply

Matrixmultiply

2

2

1

1

Q

R

uOp

Divide

Inv

Product 6

Product 5

Product 4

Product 3Product 2

Transpose 1

Transpose

++

+

++

+

minus

minus


ROV_linear_Discrete


Pk(minus)

HkPk(minus)

Pkminus1(+)

Pk(+)

++lowast

1

z

1

z

KkHkPk(minus)

kminus1(+)x

k(minus)x k(+)x

uT

uT

HTk

HkPk(minus)HTk

Hkzkzk



kminus1

Pk(minus)HTk

zk

Kk

ΦTkminus1

MATLABreg Fcn


120595(t

)


t (s)50403020100

t (s)50403020100

t (s)50403020100

0

2

4

x(t

)

minus2

0

2

y(t

)

minus1

0

1




6 Conclusions




120595(t

)

PID


8060 10020 400t (s)

0

2

4

6

8

0

5

10

15

20

25y

(t)

0

5

10

15

20

25

x(t

)

40 80 10020 600t (s)

20 40 60 80 1000t (s)








minus40

minus20

0

20

8060 10020 400t (s)

VT2

minus20

0

20

40

80 1000 4020 60t (s)


VT4

80 1000 4020 60t (s)

minus4

minus2

0

2

4


VT3

minus2

minus1

0

1

2

80 1000 4020 60t (s)


Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Sensors


B+

minus

+


uJint int

C + DJT(middot)

g(middot)

g(middot)

Tdagger120578p 120578d 120578

times

^




29 kgm2

119868119910119910

25 kgm2

119868119911119911

30 kgm2

119868119909119910


119868119909119911


119868119910119911



[119909119892 119910119892 119911119892] [0 0 0]m

[119909119887 119910119887 119911119887] [17 18 68] times 10

minus3m119883

65 kg119884V 598 kg119885

598 kg119870

0 kgm2

119872

22 kgm2

119873119903

22 kgm2

119883119906|119906|


minus1008 kgm119870119901|119901|

minus4003 kgm2

119872119902|119902|

minus1008 kgm2

119873119903|119903|

minus1008 kgm2



119866119898=

Velmax119881nom

(45)


119898

0

200

400

600

800

1000

1200

120596(t)

(rpm

)

005 01 015 020t (s)






119879and 119870


[37] given by

119870119879= sum

119904119905119906V119862119879

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (46)

119870119876= sum

119904119905119906V119862119876

119904119905119906V (119869)119904(

119875

119863

)

119905

(

119860119864

119860119900

) (119911)V (47)







Pitch ratio 11



KT

10KQ

0

01

02

03

04

05

06

07

08


Wag

enin

gen

B-se

ries p

rope

ller



Figure 7 shows 119870119879and 119870









1198794 [011 0 0]m 1205872 0

011

0025

022 022

031

Y0

Z0 X0

X0






H119896= [I6times6 0

6times6] (48)



K119901= diag 10 10 10 0 0 10

K119894= diag 001 001 001 0 0 001

K119889= diag 50 50 50 0 0 50

(49)


Corrector equations




MATLAB Fcn

Matrixmultiply

Matrixmultiply

Matrixmultiply


multiply

Matrixmultiply

Matrixmultiply

2

2

1

1

Q

R

uOp

Divide

Inv

Product 6

Product 5

Product 4

Product 3Product 2

Transpose 1

Transpose

++

+

++

+

minus

minus


ROV_linear_Discrete


Pk(minus)

HkPk(minus)

Pkminus1(+)

Pk(+)

++lowast

1

z

1

z

KkHkPk(minus)

kminus1(+)x

k(minus)x k(+)x

uT

uT

HTk

HkPk(minus)HTk

Hkzkzk



kminus1

Pk(minus)HTk

zk

Kk

ΦTkminus1

MATLABreg Fcn


120595(t

)


t (s)50403020100

t (s)50403020100

t (s)50403020100

0

2

4

x(t

)

minus2

0

2

y(t

)

minus1

0

1




6 Conclusions




120595(t

)

PID


8060 10020 400t (s)

0

2

4

6

8

0

5

10

15

20

25y

(t)

0

5

10

15

20

25

x(t

)

40 80 10020 600t (s)

20 40 60 80 1000t (s)








minus40

minus20

0

20

8060 10020 400t (s)

VT2

minus20

0

20

40

80 1000 4020 60t (s)


VT4

80 1000 4020 60t (s)

minus4

minus2

0

2

4


VT3

minus2

minus1

0

1

2

80 1000 4020 60t (s)


Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















Pitch ratio 11



KT

10KQ

0

01

02

03

04

05

06

07

08


Wag

enin

gen

B-se

ries p

rope

ller



Figure 7 shows 119870119879and 119870









1198794 [011 0 0]m 1205872 0

011

0025

022 022

031

Y0

Z0 X0

X0






H119896= [I6times6 0

6times6] (48)



K119901= diag 10 10 10 0 0 10

K119894= diag 001 001 001 0 0 001

K119889= diag 50 50 50 0 0 50

(49)


Corrector equations




MATLAB Fcn

Matrixmultiply

Matrixmultiply

Matrixmultiply


multiply

Matrixmultiply

Matrixmultiply

2

2

1

1

Q

R

uOp

Divide

Inv

Product 6

Product 5

Product 4

Product 3Product 2

Transpose 1

Transpose

++

+

++

+

minus

minus


ROV_linear_Discrete


Pk(minus)

HkPk(minus)

Pkminus1(+)

Pk(+)

++lowast

1

z

1

z

KkHkPk(minus)

kminus1(+)x

k(minus)x k(+)x

uT

uT

HTk

HkPk(minus)HTk

Hkzkzk



kminus1

Pk(minus)HTk

zk

Kk

ΦTkminus1

MATLABreg Fcn


120595(t

)


t (s)50403020100

t (s)50403020100

t (s)50403020100

0

2

4

x(t

)

minus2

0

2

y(t

)

minus1

0

1




6 Conclusions




120595(t

)

PID


8060 10020 400t (s)

0

2

4

6

8

0

5

10

15

20

25y

(t)

0

5

10

15

20

25

x(t

)

40 80 10020 600t (s)

20 40 60 80 1000t (s)








minus40

minus20

0

20

8060 10020 400t (s)

VT2

minus20

0

20

40

80 1000 4020 60t (s)


VT4

80 1000 4020 60t (s)

minus4

minus2

0

2

4


VT3

minus2

minus1

0

1

2

80 1000 4020 60t (s)


Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Corrector equations




MATLAB Fcn

Matrixmultiply

Matrixmultiply

Matrixmultiply


multiply

Matrixmultiply

Matrixmultiply

2

2

1

1

Q

R

uOp

Divide

Inv

Product 6

Product 5

Product 4

Product 3Product 2

Transpose 1

Transpose

++

+

++

+

minus

minus


ROV_linear_Discrete


Pk(minus)

HkPk(minus)

Pkminus1(+)

Pk(+)

++lowast

1

z

1

z

KkHkPk(minus)

kminus1(+)x

k(minus)x k(+)x

uT

uT

HTk

HkPk(minus)HTk

Hkzkzk



kminus1

Pk(minus)HTk

zk

Kk

ΦTkminus1

MATLABreg Fcn


120595(t

)


t (s)50403020100

t (s)50403020100

t (s)50403020100

0

2

4

x(t

)

minus2

0

2

y(t

)

minus1

0

1




6 Conclusions




120595(t

)

PID


8060 10020 400t (s)

0

2

4

6

8

0

5

10

15

20

25y

(t)

0

5

10

15

20

25

x(t

)

40 80 10020 600t (s)

20 40 60 80 1000t (s)








minus40

minus20

0

20

8060 10020 400t (s)

VT2

minus20

0

20

40

80 1000 4020 60t (s)


VT4

80 1000 4020 60t (s)

minus4

minus2

0

2

4


VT3

minus2

minus1

0

1

2

80 1000 4020 60t (s)


Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120595(t

)

PID


8060 10020 400t (s)

0

2

4

6

8

0

5

10

15

20

25y

(t)

0

5

10

15

20

25

x(t

)

40 80 10020 600t (s)

20 40 60 80 1000t (s)








minus40

minus20

0

20

8060 10020 400t (s)

VT2

minus20

0

20

40

80 1000 4020 60t (s)


VT4

80 1000 4020 60t (s)

minus4

minus2

0

2

4


VT3

minus2

minus1

0

1

2

80 1000 4020 60t (s)


Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus40

minus20

0

20

8060 10020 400t (s)

VT2

minus20

0

20

40

80 1000 4020 60t (s)


VT4

80 1000 4020 60t (s)

minus4

minus2

0

2

4


VT3

minus2

minus1

0

1

2

80 1000 4020 60t (s)


Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article development of a low-level control system...

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